where \(\hat{b}_{n, k^*}\)  is the \(n\) th element of \(\hat{b}_{k^*}\) , and \(b_n^\text{gt}\)  is the corresponding ground truth value.
As discussed in [1]}, this loss imposes a Laplacian prior on the learned distribution.
Next, we update the mixture weights \(\left\lbrace  \alpha _k \right\rbrace _{k=1}^K\)  again using the multi-class cross entropy loss \(\mathcal {L}_{\text{mix}}\) , where the positive label corresponds to the \(k^*\)  component.
The idea of only updating the prediction that best matches the ground truth was originally proposed in [2]} as the hindsight loss for multiple choice learning.
The regression loss for the entire image is defined as follows:
\(\mathcal {L}_{\text{reg}} = \frac{1}{N} \sum _i \frac{\mathcal {L}_{\text{box}, i} + \lambda \mathcal {L}_{\text{mix}, i}}{n_i}\) 
